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A331947
Factors k > 1 such that the polynomial k*x^2 - 1 produces a record of its Hardy-Littlewood constant.
6
2, 12, 20, 68, 90, 98, 132, 252, 318, 362, 398, 1722, 259668, 315180, 452042
OFFSET
1,1
COMMENTS
a(16) > 710000.
See A331940 for more information on the Hardy-Littlewood constant. The polynomials described by this sequence have an increasing rate of generating primes.
The following table provides the record values of the Hardy-Littlewood constant C, together with the number of primes np generated by the polynomial P(x) = a(n)*x^2 - 1 for 2 <= x <= r = 10^8 and the actual ratio np*(P(r)/r)/Integral_{x=2..P(r)} 1/log(x) dx.
a(n) C np C from ratio
2 3.70011 10448345 3.81422
12 4.15027 11154934 4.27219
20 4.43326 11753085 4.56136
68 5.01601 12883801 5.15797
.. ....... ........ .......
315180 7.82318 16502584 8.00057
452042 7.85323 16434699 8.02696
REFERENCES
Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
LINKS
Karim Belabas, Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Henri Cohen, High precision computation of Hardy-Littlewood constants, preprint, 1998. [pdf copy, with permission]
KEYWORD
nonn,more,hard
AUTHOR
Hugo Pfoertner, Feb 10 2020
STATUS
approved